Perhaps, our most important motivation behind studying Toeplitz+Hankel determinants is to study the large \(n\)  asymptotics of the eigenvalues of the Hankel matrix \(H_n[w]\)  associated to the symbol \(w\) . Specifically, we want
to extend the
recent results [1]} concerning the spectral asymptotics of the Toeplitz matricesThe large \(n\)  behavior of the individual
eigenvalues of Toeplitz matrices has been also addressed in a number of works – see [2]} and references therein. to the Hankel case.
The key feature which allows an effective asymptotic spectral analysis of Toeplitz matrices and, in particular, the use of the Riemann–Hilbert method, is that the characteristic polynomial of a Toeplitz
matrix is again a Toeplitz determinant with a symbol of general Fisher–Hartwig type (i.e., no conditions on the \(\beta \) -parameters).
The asymptotics of such Toeplitz determinants is given by the Basor–Tracy formula
(first conjectured by E. Basor and C. Tracy and then proved in [3]}).
However, in the case of Hankel matrices, and this is the crux of the matter,
their characteristic polynomials are not Hankel determinants. Indeed, the characteristic polynomial \(\det (H_n[w]-\lambda I)\)  of the Hankel matrix \(H_n[w]\)  is a particular Toeplitz+Hankel determinant, with \(\phi (z)\equiv -\lambda \) .
Clearly in the case of characteristic polynomial of a Hankel determinant, there is no relationship between \(\phi \)  and \(w\) , so to study the asymptotics of this determinant, one can not refer to the works [3]} or [5]} mentioned above. Here again we are directed to a methodological issue which has to be addressed at a fundamental level by formulation of a suitable Riemann–Hilbert problem.
